Label the words of this 30,60,90 right triangle SL,2SL,and SLsqrt2 01510201131.png

Question
Label the words of this 30,60,90 right triangle $$\displaystyle{S}{L},{2}{S}{L},{\quad\text{and}\quad}{S}{L}\sqrt{{2}}$$

2020-12-30
From the right triangle ABC, we have
$$\displaystyle\therefore{\sin{{\left({A}\right)}}}=\frac{{{B}{C}}}{{{A}{C}}}$$
$$\displaystyle\Rightarrow{\sin{{\left({30}^{\circ}\right)}}}=\frac{{v}}{{6}}$$
$$\displaystyle\Rightarrow{1}=\frac{{v}}{{3}}$$
$$\displaystyle\Rightarrow{v}={3}$$
From the right triangle ABC, we have
$$\displaystyle\therefore{\cos{{\left({A}\right)}}}=\frac{{{A}{B}}}{{{A}{C}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({30}^{\circ}\right)}}}=\frac{{u}}{{6}}$$
$$\displaystyle\Rightarrow\frac{\sqrt{{3}}}{{2}}=\frac{{u}}{{6}}$$
$$\displaystyle\Rightarrow\sqrt{{3}}=\frac{{u}}{{3}}$$
$$\displaystyle\Rightarrow{u}={3}\sqrt{{3}}$$
Since, the sum of all angles of a triangle is $$\displaystyle{180}^{\circ}$$
$$\displaystyle\therefore\angle{A}+\angle{B}+\angle{C}={180}^{\circ}$$
$$\displaystyle\Rightarrow{30}^{\circ}+{90}^{\circ}+\angle{C}={180}^{\circ}$$
$$\displaystyle\Rightarrow{120}^{\circ}+\angle{C}={180}^{\circ}$$
$$\displaystyle\Rightarrow\angle{C}={60}^{\circ}$$

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Label the words of this 30,60,90 right triangle $$\displaystyle{S}{L},{2}{S}{L},{\quad\text{and}\quad}{S}{L}\sqrt{{2}}$$
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